2009/06/01 by Pierre Le Doussal · 1 citation
Mathematics · Physics and Astronomy · #Anomalous diffusion #Diffusion #Exponent #Geometry #Mathematical physics #Mathematics #Physics #Power law #Probability density function #Quantum mechanics #Random Matrices and Applications #Random walk #Renormalization #Renormalization group #Saddle point #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Universality (dynamical systems) #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1088/1742-5468/2009/07/p07032
21 pages, 2 figures
arxiv created 2009/06/01 · openalex publication_date 2009/07/20 · arxiv updated 2015/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We study the Sinai model for the diffusion of a particle in a one-dimensional random potential in the presence of a small concentration ρ of perfect absorbers using the asymptotically exact real space renormalization method. We compute the survival probability, the averaged diffusion front and return probability, the two-particle meeting probability, the distribution of total distance traveled before absorption and the averaged Green's function of the associated Schrödinger operator. Our work confirms some recent results of Texier and Hagendorf obtained by Dyson–Schmidt methods, and extends them to other observables and the presence of a drift. In particular the power law density of states is found to hold in all cases. Irrespective of the drift, the asymptotic rescaled diffusion front of surviving particles is found to be a symmetric step distribution, uniform for , where ξ( t ) is a new length scale for survival ( in the absence of drift). Survival outside this sharp region is found to decay with a larger exponent, continuously varying with the rescaled distance x /ξ( t ). A simple physical picture based on a saddle point is given, and universality is discussed.